Statistical Nuances in BAO Analysis: Likelihood Formulations and Non-Gaussianities

We present a systematic comparison of statistical approaches to Baryon Acoustic Oscillation (BAO) analysis using DESI DR2 data. We evaluate four methods for handling the nuisance parameter $\beta=1/(H_0 r_d)$: marginalization, profiling, Taylor expansion, and full likelihood analysis across multiple cosmological models. Our results demonstrate that while these methods yield consistent constraints for $\Lambda$CDM and $\Omega_K$CDM models, they produce notable differences for models with dynamical dark energy parameters. Through eigenvalue decomposition of Fisher matrices, we identify extreme parameter degeneracies in $ww_a$CDM and $\Omega_Kww_a$CDM models that explain these statistical sensitivities. Surprisingly, $\Omega_K$CDM shows the highest information content across datasets, suggesting BAO measurements are particularly informative about spatial curvature. We further use skewness and kurtosis analysis to identify deviations from Gaussianity, highlighting limitations in Fisher approximations in the dark energy models. Our analysis demonstrates the importance of careful statistical treatment when extracting cosmological constraints from increasingly precise measurements.